How to combine these radicals √27- √3

To combine the radicals $\sqrt{27} - \sqrt{3}$, follow these steps:

Step 1: Simplify Each Radical Separately

The radical $\sqrt{27}$ can be simplified as follows:

$$\sqrt{27} = \sqrt{9 \times 3} = \sqrt{9} \times \sqrt{3} = 3\sqrt{3}$$

So, we have:

$$\sqrt{27} = 3\sqrt{3}$$

The radical $\sqrt{3}$ is already in its simplest form.

Step 2: Substitute the Simplified Forms

Next, substitute the simplified form of $\sqrt{27}$ back into the expression:

$$3\sqrt{3} - \sqrt{3}$$

Step 3: Combine Like Terms

Now, combine the like terms. Since both terms have the same radical part ($\sqrt{3}$), you can factor out $\sqrt{3}$:

$$3\sqrt{3} - \sqrt{3} = (3 - 1)\sqrt{3} = 2\sqrt{3}$$

So, the expression $\sqrt{27} - \sqrt{3}$ simplifies to:

$$2\sqrt{3}$$

Final Result

The final combined form of the radicals $\sqrt{27} - \sqrt{3}$ is $2\sqrt{3}$.

Radical Simplification in Math

Radical simplification involves reducing expressions that contain radicals (also known as roots) to their simplest form. This often means rewriting the expression so that the radical no longer contains a fraction or several terms.

Basic Principles

• Simplifying square roots: Combine and reduce factors inside the root to their simplest form.
• Rationalizing the denominator: Ensure there are no radicals in the denominator of a fraction by multiplying the numerator and the denominator by a conjugate if necessary.

Simplifying Square Roots

To simplify a square root:

1. Factorize the number inside the square root into its prime factors.
2. Pair up the prime factors.
3. Move one of each pair out of the square root.

For example:

$$\sqrt{72} = \sqrt{2 \cdot 2 \cdot 2 \cdot 3 \cdot 3} = 6\sqrt{2}$$

Rationalizing the Denominator

If you have a fraction with a radical in the denominator, you can remove the radical by multiplying both the numerator and the denominator by the radical itself or its conjugate.

For instance, to rationalize:

$$\frac{1}{\sqrt{2}}$$

Multiply by:

$$\frac{\sqrt{2}}{\sqrt{2}}$$

Thus:

$$\frac{1 \cdot \sqrt{2}}{\sqrt{2} \cdot \sqrt{2}} = \frac{\sqrt{2}}{2}$$

Simplifying Radicals with Variables

Radicals can also involve variables. The process is similar:

$$\sqrt{x^4 y^2} = x^2 y$$

since $x^4 = (x^2)^2$ and $y^2 = (y)^2$.

Multi-step Simplification

Complex expressions may require multiple steps. Consider:

$$\sqrt{50x^4 y^3}$$

First, break it down:

$$\sqrt{50} \cdot \sqrt{x^4} \cdot \sqrt{y^3}$$

Simplify inside:

$$5\sqrt{2} \cdot x^2 \cdot y \sqrt{y} = 5x^2 y \sqrt{2y}$$

By following these principles and steps, you can achieve radical simplification, making expressions easier to work with and understand.

Introduction

Combining like terms is a fundamental process in algebra that simplifies expressions. Like terms are terms that have the same variables raised to the same power, even if they have different coefficients.

Identifying Like Terms

To combine like terms, you first identify the terms that have the same variables and exponents. For example, in the expression:

$$3x^2 + 5x - 2x + 4 - x^2$$

The like terms are:

• $3x^2$ and $-x^2$ (both have the variable $x$ raised to the power of 2)
• $5x$ and $-2x$ (both have the variable $x$ raised to the power of 1)

Combining the Terms

1. Group the like terms together.
2. Add or subtract their coefficients.

For the expression given above, we can group and then combine like terms:

$$3x^2 - x^2 + 5x - 2x + 4$$

Combining the coefficients of the like terms, we get:

$$3x^2 - x^2 = 2x^2$$ $$5x - 2x = 3x$$

So, the simplified expression is:

$$2x^2 + 3x + 4$$

Special Cases

Constants, or terms without variables, are also considered like terms with each other. For instance, in the expression:

$$7 + 3x + 5 - 2x$$

The constants $7$ and $5$ can be combined:

$$7 + 5 = 12$$

Resulting in the simplified form:

$$3x - 2x + 12$$ $$x + 12$$

Benefits

Combining like terms:

• Simplifies expressions, making them easier to work with.
• Reduces complexity, especially useful in solving equations or inequalities.

Example

Let's look at another example:

$$4y + 7 - 5y + 3 + 2y$$

First, group the like terms:

$$(4y - 5y + 2y) + (7 + 3)$$

Now, combine the coefficients:

$$(4y - 5y + 2y) = (4 - 5 + 2)y = 1y = y$$ $$7 + 3 = 10$$

The simplified expression is:

$$y + 10$$

Being proficient at combining like terms is essential for simplifying algebraic expressions and solving algebra problems effectively.